The main purpose of this paper is to study the asymptotic stability of the combination of viscous contact wave with rarefaction waves to the Cauchy problem (REF ), (REF ) and (REF ) for the large time behavior.
We also notice that the one-dimensional Navier-Stokes-Maxwell equations \((\ref {lagrange})\)  is a combination of the compressible non-isentropic Navier-Stokes equations and the Maxwell equations through the Lorentz force.
When omitting the electrodynamic effects, the system \((\ref {lagrange})\)  reduces to the classical Navier-Stokes equations.
In fact, there are extensive studies concerning those results for the Navier-Stokes equations in the context of gas dynamics.
It is well known that the large-time behavior of solutions to the Cauchy problem for Navier-Stokes equations can converge to the Riemann solutions for the corresponding Euler equations.
As for the Riemann problem, under a proper assumption on the flux function, there are three kinds of elementary wave solutions: shock wave, rarefaction wave and contact discontinuity, and then the Riemann solution generally forms a multi-wave pattern given by a various linear combination of these three elementary waves (cf. [1]}, [2]} etc.).
Here, we mention several works on the asymptotic stability analysis of viscous wave pattern for the Navier-Stokes equations:
[3]}, [4]}, [5]} for the shock wave;
[6]}, [7]}, [8]}, [9]}, [10]} for the rarefaction wave;
[11]}, [12]}, [13]}, [14]} for the contact discontinuity.
In particular, the asymptotics toward two rarefaction waves was established in Kawashima-Matsumura-Nishihara [7]} by using an energy form associated with the physical total energy and a monotone property of the smooth approximate rarefaction waves.
Moreover, motivated by the theory of viscous shock waves [4]},
Huang-Xin-Yang [13]} succeeded in using the anti-derivative method to obtain not only the stability but also the convergence rate of the viscous contact wave for the Navier-Stokes equations and the Boltzmann equation.
However, their method is not available for the stability of the composite wave of viscous contact wave with rarefaction waves,
since the anti-derivative method is invalid for the stability of the rarefaction wave.
We should mention that the stability of the superposition of several wave patterns is more complicated and challenging
due to the fact that the stability analysis essentially depends on the underlying properties of basic wave pattern and the wave interaction between different families of wave patterns is complicated.
Recently, Huang-Matsumura in [18]} showed that the composite wave of two viscous shock waves for the Navier-Stokes equations is asymptotically stable without the zero initial mass condition.
Later Huang-Li-Matsumura in [19]} proved the stability of the combination of viscous contact wave with rarefaction waves by deriving a new estimate on the heat kernel.
This new technique was needed by Zeng in [20]} to obtain the stability of the superposition of shock waves with contact discontinuities for systems of viscous conservation laws,
and was also used by Huang-Wang in [21]} for the global stability of the same wave patterns and system as in [19]}.
In addition, some results about the asymptotic stability of the composite wave as in [19]} were also shown for some more complex models, and we refer interested readers to [24]}, [25]}, [26]}, [27]}, [28]}, and references therein.
Due to the frameworks of viscous shock wave and rarefaction wave are not compatible with each other,
the time-asymptotic stability of the combination of viscous shock wave and rarefaction wave is still an interesting and challenging question for the Navier-Stokes equations!
Recently, Moon-Jin Kang, Alexis F. Vasseur and Yi Wang have submitted the paper [29]} titled “Time-asymptotic stability of composite waves of viscous shock and rarefaction for barotropic Navier-Stokes equation” to the online database: arXiv.org.
Maybe they have made some progress on the time asymptotic stability of the combination of viscous shock wave and rarefaction wave for the Navier-Stokes equations.
